A Short Evaluation of Mr. Whiting’s Lesson Plan: Slopes and Rate-of-Change

By: Julien Milcent & Bryan Taylor

Professor Mueller

North Central College

Educational Psychology 205

14^th February, 2011

When writing a lesson plan, there are hundreds if not thousands of ways to try an get the students to learn. But, all of these ways can be put into two very distinct categories, those that follow Piaget’s method and those who follow Vygotsky’s method. In the field of psychology, these men tried to find reasonable explanations on human development and learning. Mr. Whiting , the creator of the lesson plan we evaluated, may or may not have understood what these to men’s theories were, but his lesson plan does take a Vygotsky –like approach. And we cannot talk about Vygotsky without talking about his theory that is the zone of proximal development.

The zone of proximal development is described as the idea that a learner can grasp a behavior that is right above the level at which they are at, in terms of ability. The idea being that the level right above the learners’ current level is achievable, meaning that it will challenge the individual in some way. The problem with this is that if the information is too high above the learner’s current level, the lesson will never reach the students. Mr. Whiting though takes care of this problem exceptionally. Because this lesson is about the real life exploration of the slope formula, there are a few things that students need to know before going into the lesson. Having mathematical backgrounds, the authors, Julien Milcent and Bryan Taylor, agee that you need to know the slope formula and point-slope formulas. Mr. Whiting knows this, and during his motivational activity writes, “Remind students of the slope formula, slope-intercept formula- and point-slope formula” (Whiting). Besides the fact that the students have the prior knowledge for understanding the objectives, the student must still need help.

Most students need more than one lesson for a certain topic to sink in. Therefore, from the looks of it, is the second day for the students, in terms of learning about slope, so the students will still need Mr. Whiting’s help in trying to comprehend the information. This explains why Mr. Whiting, in his lesson plan, reviews everything at the beginning of the lesson, in order to give the students another chance to see the information and understand it. He can then go into another example during the “SEE-SAY-DO” section of the lesson plan, and then go right into the applications. During the applications part he lets the students brainstorm, handing the control of the learning over to the students, where the students can work with the information by themselves. From this reasoning, Mr. Whiting’s lesson plan indeed fits Vygotsky’s zone of proximal development, as it is a stretch of the learner’s current abilities, and they know the prior knowledge for the lesson plan. So if taught correctly, this lesson plan will push the learners to be able to understand a new concept, slopes in the real world, as well as retaining the information to place it in long term memory.

Another type of learning comes through rehearsal, which helps information processing. Rehearsal is the process of transferring information from short term memory to long term memory through making connections, forming imagery, and giving meaning to material. The two main types of rehearsal are maintenance and elaboration. Maintenance is repetition of the material in order to put it into long term memory. Maintenance is seen in Mr. Whiting’s lesson plan with the repetition of the idea about the slope of lines. The students worked with the slope formula, slope-intercept, and point slope formula in the first exercise, according to Mr. Whiting’s lesson plan, about the rate of calories burning. The students then worked specifically with the slope-intercept formula in discussing whether or not it was like a recipe. Next, in the “SEE-SAY-DO” activity Mr. Whiting has the students reinforce the formulas by repeating their definitions after Mr. Whiting reads them aloud. In the review activity the students use the various formulas they have learned, by applying them to real world examples. All of these examples show repetition through the numerous occasions in which the students are encouraged to learn the formulas, allowing them to make connections, form imagery, and place meaning in them.

The second type of rehearsal is elaboration. Elaboration is working with information to make connections. Elaboration is seen in many ways in this lesson plan. The first instance of this is in the first activity involving the use of the various formulas and applying them to the burning of calories through jogging. This application of the various formulas allows the students to make connections through real world examples, such as running. Another instance of elaboration is seen when the students are asked to compare the slope-intercept formula to a recipe. This allows the students to make real world connections and places meaning to the formula. Furthermore, Mr. Whiting’s lesson plan allows his students to further make connections and associations furthering the likelihood the material would be retained, although there are still problem that can arise in that process.

Now that the lesson is taught, the students can do the problems, and everything is fine, or so they thought. The information about the real life examples of the slope formula could be in their minds at the end of the lesson, but that doesn’t mean that the students will understand it the next day, or next week on a test. There are two things that can go wrong in the process of information retrieval. One, the information may never have gotten stored. Two, the learner is just not able to retrieve it. There are different ways that each of these two problems can arise. One, is that the information could possibly never have gotten stored, which is called the cognitive load theory. This theory revolves around the fact that a learner’s memory only has a limited number of spots until it is at full capacity, and if the student’s memory is at full capacity, learning will be ineffective. In Mr. Whiting’s lesson plan, this is not an issue. This is due to the fact that Mr. Whiting has the students, “Read the sentence…” aloud idea in the SEE-SAY-DO part of his lesson plan. By having the students copy what he is saying it insures that they aren’t focused on other things going on thus forcing them to pay attention through the entire lesson. Though this method may have some success, there is a better way to try and prevent the information from overloading the student’s memory. For instance, instead of repeating everything that Mr. Whiting says, which, frankly, is a poor method choice for high school students, he could try and not go over everything they have covered recently, which was the, “slope formula, slope-intercept formula, and point-slope formula” from the Modeling Activity. Going over these methods again may be enough to throw off the students for the rest of the lesson.

Another thing that Mr. Whiting could do better is trying to combat interference. Interference is when forgetting occurs because the recall of some material interferes with the recall of some other material. The problem with this theory is that it is very hard to avoid in Math classrooms, being that so much of the information is very similar, and always builds on each other. Again, when he reviews the, “slope formula, slope-intercept formula, and point-slope formula” he is taking a risk. All three of these topics are very similar, and any time you bring close items together you run the risk of confusing the students, even in a review setting. One thing that he could have done is space the review of each topic more. Fifteen minutes is too short of a time to go over three very similar topics that differ only slightly. Mr. Whiting show plan separate lesson plans reviewing these topics individually. This could distinguish these concepts more, if Mr. Whiting taught these topics individually, by spending time each day explain who each topics differs from each other. Doing all of this in one is not an effective in helping the students spot the differences. This could be done instead of throwing the review, and then began to teach a whole new topic, being, “explor[ing] real world situations as it relates to slopes”. The one thing that is good is the Mr. Whiting goes over the slope-formula in great detail, but fails to go over the other two formulas at all. All that being said, there is one more theory of memory that Mr. Whiting does nothing to confront: cue theory.

Cue theory is when the students don’t make enough good connections to the material, and therefore do not remember the material. In this case this lesson plan is horrible. Out of the 90 minute lecture there is only 10 minutes set aside for brainstorming, and that is where the students can make the critical connections necessary for memory. For the other 80 minutes the students are in lecture mode and are just trying to take in all the information that is thrown at them. Mr. Whiting would be wise to stop having the students repeat what he says, maintenance, and focus more on the connections, elaboration. The way the lesson plan organized, there is not enough time to form the connections needed to truly learn the material and students are much more likely to go into the next class and forget all they have just learned. Next, we will discuss how they students actually learned the material.

There are two very distinct versions of learning: deductive and inductive learning. Though Mr. Whiting’s lesson plan contains opportunity for both types of learning, there is still room for improvement. The bulk of this lesson is taught in a deductive way, when Mr. Whiting gives the students the formulas and they can then remember the difference between the formulas and examples. Only at the end, in the review section, do the students by, “brainstorm other situations, such as airplanes flight landing or takeoffs…and determine the steepness of slopes and rates of change by viewing and recording [that] data”, does inductive learning occur. It is inductive learning that would take pleace because the students have to think of real world examples of slope. In school we are used to deductive learning, where a teacher just gives us rules followed by examples. The problem with this is that if we get the examples first, and then discover the rule we can form connections because we ourselves figured out the rule by ourselves. This only occurs in the last 10 minutes, and this is supposed to be the second measureable objective from the lesson plan, not nearly enough time to go over this topic. This lesson plan should have been divided between two different days, where deductive teaching would have worked when trying to go over the different formulas, because it would be too hard to have the students figure out the formulas themselves, followed by the second day where the students can inductively learn about the real life applications to slope. This would provide the students not only with real life applications, but also give them a chance to truly master slope formula by themselves, and figure out the slopes of say, “[an] airplane flight landings or takeoffs”. Subsequently, we will discover learning through imitation, otherwise known as modeling.

A good lesson plan should include modeling, a type of learning. There are two types of modeling-- one is through the teacher and the other is peer-to-peer. Teacher modeling has three distinct possible elements: consistency, different means of relaying information, and verbal mediation. Peer-to-peer modeling is seen when students work together on a certain topic. This increases self efficacy for both the tutor and the tutee, and will be discussed later in further detail.

Teacher modeling is seen this lesson plan in many ways. The first part of modeling is consistency. This lesson plan is good because Mr. Whiting administered the material in a couple of different ways while keeping the material the same. Mr. Whiting has the first exercise about jogging which asks the students to relate to variables on the same plot, come up with an equation, and analyze the slope of such equations. Next, Mr. Whiting asks the class to discuss how the slope-intercept form is like or unlike a recipe. Then, Mr. Whiting makes it a little more hands on asking them to repeat important facts and then find the slopes of lines when given to points. Lastly, Mr. Whiting asks the class to review the lesson of slopes through discussion. Through all of this Mr. Whiting is using different means to get the information to his students. The first example is a personal story, something easily relatable, a good way to open a lesson. Mr. Whiting then uses various methods to present the information: discussion for auditory learners, putting problems on the board for visual learners and so students can imitate him (a form of modeling), and again wraps up the lesson discussion which helps the students to summarize what they have learned. While Mr. Whiting provided different ways to look at the information, the information itself was the same in each example therefore we say that Mr. Whiting’s lesson plan has consistency.

Something that Mr. Whiting does not have in the lesson plan is peer-to-peer modeling. Peer-to-peer modeling occurs when students working together learn or review topics. This strategy tends to last longer than just straight lecture from the teacher. Working with peers increases self-efficacy, which is the notion that you think you can be successful at a task. This is because the students will see their peers are either also struggling with a task or also accomplishing tasks which, either way, reinforces the idea that they are not alone and that the material is learnable. In this lesson plan we can see peer-to-peer modeling in discussion but only to an extent. Mr. Whiting still leads the discussion, so the students are not fully working one on one and the idea of raising self efficacy doesn’t exist. This lesson plan could have been greatly improved if the students were asked to break up into groups and practice a portion of the material covered in this lesson, possible with the help of a tutor.

In the lesson plan tutoring comes into play when he gives the students the, “five slope and rate of change problems,” at the end of the lesson, but only for the ELL students. This is a perfect opportunity gone to waste. Mr. Whiting only allows the students who are deemed in need of the extra assistance, such as ELL students, to work with a peer tutor. This is a good idea, for everyone, especially if the tutor’s a coping one, gets a chance to increase their self-efficacy by helping another peer with it, as well as make connections because they are teaching the material themselves in a way. This also increases the tutee’s self-efficacy because they are getting help from a peer, someone who was once in their shoes, which makes the learning more involved for them, and much more interactive, because of the one-on-one approach. With both the tutor and the tutee getting a chance to work together, they can both reach a higher understanding of the material. This is why Mr. Whiting should let everyone work in pairs, to allow the students who might understand the material well to get a chance and help other students in the classroom. This will allow all the students the chance to work one-on-one and work with the information together. The way that Mr. Whiting has it organized right now doesn’t do anything for the students that need some more time with the material, if anything it will make them feel dumb, because they will think that they cannot do the problems on their own. If we let the students work in pairs then with both sides working together, they will all have a high chance of understanding the problems, and working out the right answer.